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i'm trying to understand time series and lag operators. However there is a lot of stuff about it on the internet (and too on stack overflow, but not what I'm looking for), I cannot understand, how to compute Lag (sometimes called Backshift) operator from a given time series. I know, how it works ($BX_{t}=X_{t-1}$). Imagine, that we have some time series T:
$$ T = (x_{1},x_{2},x_{3},...,x_{n-1},x_{n}) $$ in time $$(t_{1},t_{2},t_{3},...,t_{n-1},t_{n})$$ Now for example you have ARMA model, where you use equation $$ \phi(B)X_{t}=θ_{0}+θ(B)a_{t} $$ where $\phi(B)=1-\phi_{1}B-...-\phi_{p}B^{p}$ and $θ(B)=1-θ_{1}B-...-θ_{q}B^{q}$ are polynomials in B with degrees p and q.

Now what I want is to compute all B values (or better polynomials $\phi(B)$ and $θ(B)$) for equation above (for simplicity imagine, that we have ARMA(1,2)). I have seen some equations and computing using lag operator, but values were already given (see here page 4). I cannot find how to find out values for lag.

I was thinking, that I can somehow use linear regression, but I'm not sure. My second idea was to compute it as here:
If $BX_{t}=X_{t-1}$, than I can compute B as B = $X_{t-1}/X_{t}$, but it cannot be same for all times $T_{s}$ same right?
For example: $B_{1} = x_{1}/x_{2}$, $B_{2}=x_{2}/x_{3}$,... but $B_{1}\ne B_{2}$, so which one to use? Or should I use both?
I was thinking, that B can be a polynom itself, but how then compute polynomials $\phi(B)$ and $θ(B)$?

Maybe I all understand it wrong.

Thank you

joettriscik
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    The question is not clear to me. Are you looking for the roots of the AR and MA polynomials? Do you want to get parameter estimates of the AR and MA coefficients? Explaining the purposes or setting where you are using the ARMA model may help to understand the question. – javlacalle Jul 03 '15 at 15:04
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    Yes, I am looking for roots (small phi and later Backshift if it is possible). If you take a look at Wiki (https://en.wikipedia.org/wiki/Autoregressive_model or moving average model too), first I want equation parameters $\phi_{i}$ and then to find out backshift parameter. I am looking for backshift parameters because I'm trying to understand this document: http://halweb.uc3m.es/esp/Personal/personas/dpena/articles/biomk00.pdf. I want to try it understand and implement simple code to see, in which cases it works the best. – joettriscik Jul 04 '15 at 15:42
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    You can get the roots of the AR polynomial by equating to zero and solving for $B$. For example, given the following AR model $x_t = 0.5y_{t-1}+a_t$, $\phi(B)$ is defined as $(1-0.5B)$, then solving for $B$ in $1-0.5B=0$ we find that the root is $B=2$. The same applies for the MA polynomial. – javlacalle Jul 04 '15 at 17:19
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    You may be interested in this document, which discusses similar concepts to the reference that you gave but in the context of univariate time series. If you can relate your question to some point in this document if may be easier to me to understand it and give you an answer. – javlacalle Jul 04 '15 at 17:19
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    Thank you, I'll study document and than I'll write another questions. But my first question is. Lets say, that you have any time series defined as times T = ($t_{1}$,$t_{2}$,...,$t_{n}$) and values V = ($v_{1}$,$v_{2}$,...,$v_{n}$) in corresponding time (value $v_{k}$ is in time $t_{k}$). What is the method to obtain model x_{t} as You wrote above? Using linear regression? I'm red it somethere, but I'm not sure, if it is the right way. – joettriscik Jul 05 '15 at 18:21
  • *I've read it somewhere... – joettriscik Jul 05 '15 at 18:27
  • An autoregressive model can be estimated as a linear regression model, by ordinary least squares. All you need to do is to create the lagged variables and apply the OLS estimator. ARMA models could be estimated as a linear regression, but it is not so straightforward (it would involve an iterative process where the residuals of a previous model are taken as an estimate of the innovations $a_t$). – javlacalle Jul 05 '15 at 19:19
  • It is more common to estimate ARMA models by maximum likelihood. Usually the model is written in state space form, then the Kalman filter can be used to evaluate the likelihood function, which is in turn maximized using a numerical optimization algorithm, e.g. BFGS. But this is not the only possible approach. The topic is too lengthy to be discussed here. – javlacalle Jul 05 '15 at 19:19
  • I would recommend you choosing a software package that fits ARMA models and read the documentation to see which method is implemented in the package. There you will probably find references to some textbook or papers. Then you will probably come up with more specific questions. – javlacalle Jul 05 '15 at 19:22
  • Yes, that is what I needed! Thank you for your answers. I will look at it and do some research. – joettriscik Jul 06 '15 at 19:48

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